Theorem pnt 24452

Description: The Prime Number Theorem: the number of prime numbers less than x tends asymptotically to x / log ( x ) as x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)

Assertion

Ref	Expression
pnt	|- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Proof of Theorem pnt

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 9642	. . . . . . 7 |- 1 e. RR
2	1	rexri 9693	. . . . . 6 |- 1 e. RR*
3		1lt2 10776	. . . . . 6 |- 1 < 2
4		df-ioo 11639	. . . . . . 7 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
5		df-ico 11641	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6		xrltletr 11454	. . . . . . 7 |- ( ( 1 e. RR* /\ 2 e. RR* /\ w e. RR* ) -> ( ( 1 < 2 /\ 2 <_ w ) -> 1 < w ) )
7	4, 5, 6	ixxss1 11653	. . . . . 6 |- ( ( 1 e. RR* /\ 1 < 2 ) -> ( 2 [,) +oo ) C_ ( 1 (,) +oo ) )
8	2, 3, 7	mp2an 678	. . . . 5 |- ( 2 [,) +oo ) C_ ( 1 (,) +oo )
9		resmpt 5154	. . . . 5 |- ( ( 2 [,) +oo ) C_ ( 1 (,) +oo ) -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
10	8, 9	mp1i 13	. . . 4 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
11	8	sseli 3428	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. ( 1 (,) +oo ) )
12		ioossre 11696	. . . . . . . . . . 11 |- ( 1 (,) +oo ) C_ RR
13	12	sseli 3428	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> x e. RR )
14	11, 13	syl 17	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
15		2re 10679	. . . . . . . . . . 11 |- 2 e. RR
16		pnfxr 11412	. . . . . . . . . . 11 |- +oo e. RR*
17		elico2 11698	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) ) )
18	15, 16, 17	mp2an 678	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) )
19	18	simp2bi 1024	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
20		chtrpcl 24102	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
21	14, 19, 20	syl2anc 667	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ )
22		0red 9644	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 e. RR )
23	1	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 e. RR )
24		0lt1 10136	. . . . . . . . . . . 12 |- 0 < 1
25	24	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 < 1 )
26		eliooord 11694	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) )
27	26	simpld 461	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 < x )
28	22, 23, 13, 25, 27	lttrd 9796	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> 0 < x )
29	13, 28	elrpd 11338	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> x e. RR+ )
30	11, 29	syl 17	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
31	21, 30	rpdivcld 11358	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / x ) e. RR+ )
32	31	adantl 468	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. RR+ )
33		ppinncl 24101	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
34	14, 19, 33	syl2anc 667	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
35	34	nnrpd 11339	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
36	13, 27	rplogcld 23578	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> ( log ` x ) e. RR+ )
37	11, 36	syl 17	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
38	35, 37	rpmulcld 11357	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
39	21, 38	rpdivcld 11358	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
40	39	adantl 468	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
41	30	ssriv 3436	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR+
42		resmpt 5154	. . . . . . . 8 |- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
43	41, 42	ax-mp 5	. . . . . . 7 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) )
44		pnt2 24451	. . . . . . . 8 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1
45		rlimres 13622	. . . . . . . 8 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1 -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
46	44, 45	mp1i 13	. . . . . . 7 |- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
47	43, 46	syl5eqbrr 4437	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ~~> r 1 )
48		chtppilim 24313	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
49	48	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
50		ax-1ne0 9608	. . . . . . 7 |- 1 =/= 0
51	50	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
52	39	rpne0d 11346	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
53	52	adantl 468	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
54	32, 40, 47, 49, 51, 53	rlimdiv 13709	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
55	14	recnd 9669	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. CC )
56		chtcl 24036	. . . . . . . . . . . . 13 |- ( x e. RR -> ( theta ` x ) e. RR )
57	13, 56	syl 17	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. RR )
58	57	recnd 9669	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. CC )
59	11, 58	syl 17	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC )
60	55, 59	mulcomd 9664	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( x x. ( theta ` x ) ) = ( ( theta ` x ) x. x ) )
61	60	oveq2d 6306	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) )
62	38	rpcnd 11343	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
63	30	rpne0d 11346	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x =/= 0 )
64	21	rpne0d 11346	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) =/= 0 )
65	62, 55, 59, 63, 64	divcan5d 10409	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
66	61, 65	eqtrd 2485	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
67	38	rpne0d 11346	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) =/= 0 )
68	59, 55, 59, 62, 63, 67, 64	divdivdivd 10430	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) )
69	34	nncnd 10625	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
70	37	rpcnd 11343	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. CC )
71	37	rpne0d 11346	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) =/= 0 )
72	69, 55, 70, 63, 71	divdiv2d 10415	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
73	66, 68, 72	3eqtr4d 2495	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
74	73	mpteq2ia 4485	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
75		1div1e1 10300	. . . . 5 |- ( 1 / 1 ) = 1
76	54, 74, 75	3brtr3g 4434	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
77	10, 76	eqbrtrd 4423	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
78		ppicl 24058	. . . . . . . . . 10 |- ( x e. RR -> (π ` x ) e. NN0 )
79	13, 78	syl 17	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. NN0 )
80	79	nn0red 10926	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. RR )
81	29, 36	rpdivcld 11358	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
82	80, 81	rerpdivcld 11369	. . . . . . 7 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. RR )
83	82	recnd 9669	. . . . . 6 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
84	83	adantl 468	. . . . 5 |- ( ( T. /\ x e. ( 1 (,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
85		eqid 2451	. . . . 5 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
86	84, 85	fmptd 6046	. . . 4 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) : ( 1 (,) +oo ) --> CC )
87	12	a1i 11	. . . 4 |- ( T. -> ( 1 (,) +oo ) C_ RR )
88	15	a1i 11	. . . 4 |- ( T. -> 2 e. RR )
89	86, 87, 88	rlimresb 13629	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 <-> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 ) )
90	77, 89	mpbird 236	. 2 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
91	90	trud 1453	1 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Syntax hints:   <-> wb 188   /\ w3a 985   = wceq 1444   T. wtru 1445   e. wcel 1887   =/= wne 2622   C_ wss 3404   class class class wbr 4402   |-> cmpt 4461   |` cres 4836   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   x. cmul 9544   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   [,)cico 11637   ~~> r crli 13549   logclog 23504   thetaccht 24017  πcppi 24020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-o1 13554  df-lo1 13555  df-sum 13753  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-em 23918  df-cht 24023  df-vma 24024  df-chp 24025  df-ppi 24026  df-mu 24027

This theorem is referenced by: (None)